It is not obvious why momentum should be conserved. Momentum is a very abstract concept, and the principle of conservation of momentum is really a consequence of other, fundamental Laws of physics:
The principle of Relativity
The Principle of Conservation of Energy
In the inertial frame O' above the particle is moving with velocity
in the y' direction, and it's kinetic energy is
and if the particle experiences no external force this kinetic energy is constant. In the inertial frame O', the velocity of the particle is
and it's kinetic energy is
and since the particle is subject to no external forces in O', it is subject to no external forces in O, so this kinetic energy is constant.
Hence the difference of the two kinetic energies is constant.
is constant and so is
thereforeis a constant vector and so is
and we can now sum over many particles to obtain